In statistical analysis we collect various data and present it in a particular format for analyzing. This data collected can be represented in different forms like graphs, tables, pie charts, histograms etc.

A Cumulative Frequency is such a format in which we tabulate the observed data in Less than or More than Format. It is obtained with the help of the frequencies of different classes in statistics.

There are different types of frequency distribution like, Grouped frequency distribution, Ungrouped frequency distribution, binomial distribution, mean and variance of binomial distribution, Cumulative frequency distribution, Relative frequency distribution, Relative cumulative frequency distribution under elementary statistics.

In this article we will discuss Cumulative Frequency in detail.

What is Cumulative Frequency?

Cumulative frequency is the running total of frequencies up to a certain point in a list of data. To calculate it, you start with the first frequency, then keep adding the next one to the total so far. For example, add the frequency of the first group to the second, then that total to the third, and so on. This gives you a cumulative frequency table, also called a cumulative frequency distribution table, which helps show how the total builds up across the data.

We use cumulative frequency to understand how many values fall below or above a certain level in a set of data.

Meaning of Cumulative Frequency

Cumulative frequency (often written as c.f.) means the total number of data points collected up to a specific point in the list. Simply put, it adds up all the frequencies from the beginning up to that point, no matter which group or class it belongs to.

There are two types of cumulative frequency:

• Less than type – shows how many values are less than or equal to a certain number.
• Greater than type – shows how many values are greater than or equal to a certain number.

These help us easily find out how many values fall below or above a particular limit in any dataset.

Cumulative Frequency Distribution

Cumulative Frequency Distribution is a summary of a set of data showing the frequency of items less than or equal to the upper class limit of each class. Cumulative frequency distribution is a form of frequency distribution that is created to represent the sum of the frequencies of a class and all classes below. If we have a frequency distribution table then we can arrange it in a special way known as Cumulative Frequency Distribution.

Let us consider that we have the following data.

Now we will construct the Cumulative Frequency Table as shown below.

In the above table we have calculated the cumulative distribution using less than sum by adding the present frequency to the next frequency.

Cumulative Frequency Formula 

Cumulative frequency tells us how many data values fall at or below a certain number. To find the cumulative frequency at any given point, we simply add up all the frequencies from the beginning up to that point.

Formula:

CFi = f₁ + f₂ + f₃ + ... + fi

Or in symbols:

CFi = ∑(from j = 1 to i) fj

What This Means:

• CFi is the cumulative frequency at the i-th value.
• fj is the frequency of the j-th value in the list.

Cumulative Frequency Distribution Types

Cumulative Frequency Distribution can be classified mainly to two types based on our way of calculation data.

• More than Cumulative Frequency Distribution, and
• Less than Cumulative Frequency Distribution,

More than Cumulative Frequency Distribution

Cumulative Frequency Distribution which has its cumulative frequency arranged in descending order is known as More than Cumulative Frequency Distribution. It is obtained by subtracting the succeeding class frequency from the present frequency. The cumulative frequency here starts from the highest number that is the total number of observations and ends at the minimum frequency. We get the final value as the frequency of the last class interval.

How to Draw a More Than Cumulative Frequency Curve (Ogive)

The image above shows a "More Than Cumulative Frequency Curve", also known as a more than ogive. This graph helps us understand how many values in a dataset are greater than or equal to a certain class limit.

Step-by-Step Instructions to Draw the More Than Ogive:

1. Prepare a Frequency Table

• Start with class intervals and their frequencies.
• Then calculate the "more than" cumulative frequencies by starting from the top and subtracting frequencies one by one.

2. Mark the Axes

• On the X-axis, mark the lower class limits of the intervals (e.g., 10, 20, 30…).
• On the Y-axis, mark the cumulative frequencies (e.g., 40, 32, 25…).

3. Plot the Points

• For each class, plot a point where:

         X = lower class limit

         Y = cumulative frequency

• For example, if the cumulative frequency is 35 at the lower limit 30, plot (30, 35).

4. Join the Points

• Connect all plotted points using a smooth curve.
• The curve should gradually slope downward, showing decreasing cumulative frequency.

5. Label Your Graph

• Title: More Than Ogive
• X-axis: Lower Class Limit
• Y-axis: Cumulative Frequency

For example, if we have the following table where the total number of observations is given as 40.

Now we will convert this into More than Cumulative Frequency Distribution.

The above distribution can be represented in a graph using More than Ogive as shown below.

Less than Cumulative Frequency Distribution

Cumulative Frequency Distribution which has its cumulative frequency arranged in ascending order is known as Less than Cumulative Frequency Distribution. It is obtained by adding the succeeding class frequency to the present frequency. The cumulative frequency here starts from the frequency of the first class interval and ends at the maximum frequency. We get the final value as the total number of observations given.

For example, We use the same example as above and create a Less than Cumulative Frequency Distribution table.

How to Draw a Less Than Cumulative Frequency Curve (Ogive)

The graph you see above is a "Less Than Cumulative Frequency Curve" (also called a less than ogive). It shows how data builds up as we move through increasing class limits. Here's how you can draw this type of curve:

Step-by-Step Instructions:

1. Create a Frequency Table

• Start with a table that includes class intervals and their frequencies.
• Then calculate the "less than" cumulative frequency for each class by adding the frequencies step by step.

2. Mark the Axes

• On the X-axis, mark the upper class limits of each interval (e.g., 10, 20, 30…).
• On the Y-axis, mark the cumulative frequencies (e.g., 5, 12, 20…).

3. Plot the Points

• For each class, plot a point where:

         X = upper class limit

          Y = cumulative frequency

• For example, if the cumulative frequency is 25 at the upper limit 30, plot the point (30, 25).

4. Join the Points

• Connect all the plotted points with a smooth curve (not a straight line).
• Do not use a ruler—the curve should flow naturally.

5. Label the Graph

      Add titles and labels:

• Title: Less Than Ogive
• X-axis: Upper Class Limit
• Y-axis: Cumulative Frequency

The above distribution can be represented in a graph using Less than Ogive as shown below.

Cumulative Frequency Polygon 

A Cumulative Frequency Polygon is a line graph that shows how many data values are less than or equal to a certain number in a dataset. It helps us understand how the data builds up across different groups or intervals.

To draw this graph, we:

• Take the upper boundary (end value) of each group or class interval.
• Plot the cumulative frequency for each of these boundaries.
• Join the plotted points using straight lines.

Types of Cumulative Frequency Polygons

There are two main types of cumulative frequency polygons:

1. Less Than Cumulative Frequency Polygon
   • Shows how many values are less than or equal to a specific value.
2. More Than Cumulative Frequency Polygon
   • Shows how many values are greater than or equal to a specific value.

Note: A Cumulative Frequency Curve is similar but is drawn using a smooth, freehand curve. In contrast, a Cumulative Frequency Polygon is made with straight lines connecting the points.

Cumulative Frequency Graph

A Cumulative Frequency Graph is a mix of a histogram and a cumulative line graph. It helps visualize both the data groups and how the data accumulates.

Here’s how you can create one step by step:

1. Organize the Data:

• Divide your values into class intervals (like 0–10, 10–20, etc.).

2. Find Cumulative Frequencies:

• Add up the frequencies from the first interval up to the current one.   Example:                                                     If the frequencies are 5, 8, 12                     → Cumulative: 5, 5+8 = 13, 13+12 = 25.

3. Draw Histogram Bars:

• Make vertical bars (like a standard histogram). The height of each bar is the frequency of that interval.

4.  Plot Cumulative Frequency Points:

• On the same graph, plot a point for each class using:

         X-axis: The upper limit of the class interval

         Y-axis: The cumulative frequency

5. Connect the Dots:

• Use straight lines to connect these points to form a Cumulative Frequency Curve (polygon or curve).

Relative Cumulative Frequency Graphs 

A Relative Cumulative Frequency Graph is a type of graph that shows how much of the total data falls below a certain value. This kind of graph is often called an Ogive. It helps us understand the percentile of data — meaning how much percentage of data is less than or equal to a specific value.

What Does This Graph Show?

• Percentile: It tells you the percentage of data below a particular point. For example, if your test score is in the 75th percentile, it means you did better than 75% of the students.
• Shape of Distribution: The curve shows how the data is spread out. This helps in comparing different values and seeing how far they are from the average.

Let’s Understand with a Simple Example

Situation: A bakery wants to see how many cakes it sold each week over one month and wants to see the percentage of total sales week by week.

Weekly Data:

Step 1: Find the Total Sales
Add all cakes sold:
8 + 15 + 12 + 10 = 45 cakes

Step 2: Find Relative Frequency (in Decimal)
Relative frequency is how much one week’s sales are compared to the total sales.

Week 1: 8 ÷ 45 = 0.18
Week 2: 15 ÷ 45 = 0.33
Week 3: 12 ÷ 45 = 0.27
Week 4: 10 ÷ 45 = 0.22

Step 3: Find Relative Cumulative Frequency

This is where we add the percentages week by week:

Note: The final value will always be 1 (or 100%) since it includes all the data.

Step 4: Graph the Data

To make the graph:

• Plot weeks on the x-axis.
• Plot Relative Cumulative Frequency on the y-axis (as decimals or percent).
• Connect the points with a smooth curve.

Uses of Cumulative Frequency Distribution

The uses of Cumulative Frequency Distribution are listed below.

• It is widely used to represent the raw data in a tabular format and hence study effectively. This tabular format helps to minimize the chances of error.
• It is used by various organizations to display statistical events like weather forecast, census, thermal graphs. It helps in accurate prediction and thereby in research.
• In different labs data is mainly represented in this format for studies and research.
• It is also used for determining median and different quartiles.

Cumulative Frequency Distribution Solved Examples

Example 1: Calculate the Less than Cumulative Frequency Distribution for the following data. Also draw the graph.

Solution:

We can calculate the Less than Cumulative Frequency Distribution by adding the successive frequencies to the present frequency.

We plot the graph as shown below.

Example 2: Consider the following frequency distribution table for a given total of 31. Calculate the More than Cumulative Frequency Distribution for the following data. Also draw the graph.

Solution:

We first determine the more than cumulative frequencies.

Now we plot the graph as shown below.

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